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Properties

Label 2-12-12.11-c11-0-2
Degree $2$
Conductor $12$
Sign $0.932 - 0.360i$
Analytic cond. $9.22011$
Root an. cond. $3.03646$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.08 − 45.2i)2-s + (−399. + 132. i)3-s + (−2.04e3 + 98.5i)4-s − 4.15e3i·5-s + (6.43e3 + 1.79e4i)6-s + 3.91e4i·7-s + (6.68e3 + 9.24e4i)8-s + (1.41e5 − 1.05e5i)9-s + (−1.87e5 + 4.51e3i)10-s + 7.15e5·11-s + (8.04e5 − 3.10e5i)12-s − 6.05e5·13-s + (1.77e6 − 4.26e4i)14-s + (5.50e5 + 1.65e6i)15-s + (4.17e6 − 4.03e5i)16-s + 5.31e6i·17-s + ⋯
L(s)  = 1  + (−0.0240 − 0.999i)2-s + (−0.949 + 0.315i)3-s + (−0.998 + 0.0481i)4-s − 0.593i·5-s + (0.337 + 0.941i)6-s + 0.880i·7-s + (0.0721 + 0.997i)8-s + (0.801 − 0.598i)9-s + (−0.593 + 0.0142i)10-s + 1.33·11-s + (0.932 − 0.360i)12-s − 0.452·13-s + (0.880 − 0.0211i)14-s + (0.187 + 0.563i)15-s + (0.995 − 0.0960i)16-s + 0.908i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.932 - 0.360i$
Analytic conductor: \(9.22011\)
Root analytic conductor: \(3.03646\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :11/2),\ 0.932 - 0.360i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.885526 + 0.165149i\)
\(L(\frac12)\) \(\approx\) \(0.885526 + 0.165149i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 + 45.2i)T \)
3 \( 1 + (399. - 132. i)T \)
good5 \( 1 + 4.15e3iT - 4.88e7T^{2} \)
7 \( 1 - 3.91e4iT - 1.97e9T^{2} \)
11 \( 1 - 7.15e5T + 2.85e11T^{2} \)
13 \( 1 + 6.05e5T + 1.79e12T^{2} \)
17 \( 1 - 5.31e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.51e7iT - 1.16e14T^{2} \)
23 \( 1 + 4.01e7T + 9.52e14T^{2} \)
29 \( 1 - 1.84e6iT - 1.22e16T^{2} \)
31 \( 1 - 2.41e8iT - 2.54e16T^{2} \)
37 \( 1 - 4.80e8T + 1.77e17T^{2} \)
41 \( 1 + 1.82e8iT - 5.50e17T^{2} \)
43 \( 1 + 1.12e9iT - 9.29e17T^{2} \)
47 \( 1 + 4.80e8T + 2.47e18T^{2} \)
53 \( 1 - 4.89e9iT - 9.26e18T^{2} \)
59 \( 1 - 3.77e9T + 3.01e19T^{2} \)
61 \( 1 - 1.72e9T + 4.35e19T^{2} \)
67 \( 1 - 1.69e10iT - 1.22e20T^{2} \)
71 \( 1 + 1.32e10T + 2.31e20T^{2} \)
73 \( 1 + 3.00e10T + 3.13e20T^{2} \)
79 \( 1 + 3.23e9iT - 7.47e20T^{2} \)
83 \( 1 + 3.04e9T + 1.28e21T^{2} \)
89 \( 1 - 3.87e10iT - 2.77e21T^{2} \)
97 \( 1 - 3.29e10T + 7.15e21T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.60478039803078274398189131468, −16.50237071798024253897453484993, −14.59256749832778833465862793183, −12.45735756367225304875791181967, −11.91042555582340522809667780153, −10.20869931564414266167162122403, −8.828185099612389341073313584470, −5.76325675225368909320721971713, −4.10835952348305585243274994718, −1.41520678555705035552153965773, 0.57119665342609943212569873329, 4.44739595473349923524677023682, 6.39895120855558323812335584042, 7.37521815190353861059602737222, 9.750551131497611316271619596106, 11.48274852352088860438419412597, 13.32782055133356308110480155409, 14.60315482470446945584804385173, 16.27509722914776845824109023449, 17.24748047070096726073894044364

Graph of the $Z$-function along the critical line